Senior Member

اسم الكريم : what's your name? ( as Andrew pointed out ), it is a phrase which has a sense of formality, used mainly by official or governmental employees , and moved to the usage in everyday life. ( The implication I get when I hear this phrase is: someone, who is always in a hurry, wants to know his addresse's name quickly , to go on a certain procedure).

Actually, I think it can be a good equivalent.
The use of the third [implied] person in the Arabic structure reminds me of the Spanish usage of "usted" where you address people formally by using the verb conjugations of the third person.
Or like the French: Votre majesté s'appelle/veut/dit/exige...

And it's not a very common form in Arabic.

I think the closest equivalent is "What is your good name?", since we also usually hear this phrase in formal settings such as in offices and government departments.

It's not always in formal settings. And the meaning or connotation can either be respectful or ironic, depending on the tone, the facial expression, the context....

In formal situations in Egypt, like in a bank for example, you'd most likely be asked: اسم حضرتك إيه؟ or simply اسم حضرتك؟ rather than "ism el karim", which is not as often used these days as it used to be.

Imperfect Mod

It's relatively common in Palestinian Arabic, but the variation I'm used to hearing is الاسم الكريم (not اسمك الكريم). I would agree with Andrew that Arrius's translation is a little excessive, but I have to say I've never heard "your good name" in that context in English (when I hear "X's good name" I think of someone's reputation). To me, it's just a polite way to ask someone what their name is, similar in politeness to something like "Could I get your name?" in English as opposed to just "What's your name?".